All Questions
46,952
questions with no upvoted or accepted answers
4
votes
0
answers
227
views
Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?
Let $G$ be a complex Chevalley group (not necessarily adjoint type) with $\operatorname{\mathbb{C}-rank}\geq2$ and let $H$ be a normal subgroup of $G(\mathbb Z)$ with a finite index (so $H$ is Zariski ...
- 71
4
votes
0
answers
113
views
Amalgamated subgroup of an HNN extension finitely generated
Baumslag proved that if $G= A \ast_{C} B$ is an amalgamated free product where $A$ and $B$ are finitely presented, $G$ is finitely presented if and only if $C$ is finitely generated.
Similarly, by ...
- 321
4
votes
0
answers
149
views
Adjunctions in a weak $2$-category
Is the notion of an adjunction well defined in an arbitrary weak $2$-category?
In particular, it seems like we need at least invertible unitors in the ambient $2$-category to write down the triangle ...
- 9,009
4
votes
0
answers
277
views
Reference for results about planar graphs
A colleague and I are writing a paper in which we need to make use of some basic facts about planar graphs. I would strongly prefer to simply give references for the results if possible, because the ...
- 2,289
4
votes
0
answers
73
views
universal 0-dimensional separable metric subspaces
Let $\ \mathscr U:=(U\ \delta)\ $ be a separable metric space which is universal for all finite metric spaces, i.e. for every finite metric space $ \mathscr X:=(X\ d)\ $ there
exists an isometric ...
- 4,686
4
votes
0
answers
173
views
Algebraic varieties associated to finite groups
Have the following equations been studied in the literature?
Let $G$ be a finite group.
Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...
- 2,768
4
votes
0
answers
151
views
Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?
Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots.
We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\...
- 8,707
4
votes
0
answers
261
views
What is the matrix?
Let $Bun_{n}(\mathbb{P}^1)$ be the stack of rank n bundles of degree zero on $\mathbb{P}^1$. Fix a finite field. I will write $[X]$ for the number of points on $X$. As usual if $X$ is a stack, we ...
- 8,643
4
votes
0
answers
86
views
"Generic member" in a nontempered L-packet
It is a standard conjecture that there is a unique generic member in a tempered L-packet. Is there an analogue of this for non-tempered ones, namely does one expect that there is a unique "most ...
- 1,014
4
votes
0
answers
114
views
Log-Sobolev Inequalities for convex bodies
For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form:
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\...
- 509
4
votes
0
answers
130
views
System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence
I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands.
Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...
- 7,521
4
votes
0
answers
109
views
A 4-manifold with a special non-free circle action?
Let $X$ be an oriented closed 4 manifold, with a nontrivial orientation-preserving circle action.
Question Is there an example such that $X/S^1$ is an orbifold (not a manifold), with a trivial first ...
- 41
4
votes
0
answers
97
views
Partially fibered categories vs T-Multicategories
Short version: This is a reference request question. I would like to know if something has been written on the connection between $T$-multicategory (for $T$ a monad on a category $\mathcal{E}$), and ...
- 40.2k
4
votes
0
answers
108
views
Model for Shimura curves
There is a list of Shimura curves (upto genus 2) in the paper https://math.dartmouth.edu/~jvoight/articles/shimbound-mcom-fixed-errata.pdf. My question is can I construct corresponding models for them ...
- 41
4
votes
0
answers
78
views
Existence of function minimising L^1 distance to a sequence of functions
Here all functions are from $[0, 1] \to \mathbb R$.
Let $f_i$ be a sequence of continuous functions such that there exists some $M > 0$ such that $|f_i| < M$ for all $i$. Does there always ...
- 2,039
4
votes
0
answers
89
views
Existence of uniformly continuous right inverse of bounded linear (surjective) maps between Banach spaces
Is there a known description of pairs of Banach spaces $(E,F)$ such that each continuous linear map from $E$ onto $F$ has a uniformly continuous right inverse? In his famous paper in Fund Math (2008), ...
- 101
4
votes
0
answers
90
views
Length of exceptional sequences
Let $A$ be a finite dimensional algebra over a field $k$. A module $M$ is called exceptional in case $End_A(M)=k$ and $Ext_A^i(M,M)=0$ for $i>0$.
A tuple $(M,N)$ is called exceptional in case $M$ ...
- 26.1k
4
votes
0
answers
157
views
Is simplicial localization part of a Quillen equivalence between relative categories and simplicial categories?
There are many models for $\infty$-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial ...
- 4,850
4
votes
0
answers
180
views
Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two
In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...
4
votes
0
answers
96
views
When are extensions of algebraically good groups algebraically good?
Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
- 1,635
4
votes
0
answers
414
views
View Dirichlet character as a character of Galois group
In Jaclyn Lang's article "On the image of the Galois representation associated to non-CM Hida family" section 2, the Dirichlet character $\chi$ module $N$ is also viewed as a character $\chi\colon\...
- 337
4
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0
answers
97
views
Is every locally compact connected homogeneous metric space a manifold cross a continuum?
Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$...
- 10.3k
4
votes
0
answers
138
views
Is well-pointedness the reason that the internal/external distinction seems not to apply to $\mathbf{Set}$?
When reasoning about the category of sets, we usually don't have to worry about the internal/external distinction. For example, if $f : X \rightarrow Y$ is a morphism of sets, then $f$ is either ...
- 3,693
4
votes
0
answers
198
views
Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 91
4
votes
0
answers
139
views
Adjoint actions in abstract tensor categories
Say we have a Lie group $G$. The category of (finite, complex) representations $\mathsf{Rep}\,G$ contains the adjoint representation $\mathfrak{g}$ which has many special properties. For instance $\...
- 275
4
votes
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answers
138
views
Are these two arguments incompatible?
I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong.
First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...
- 1,379
4
votes
0
answers
229
views
Why does $\theta: \mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p$ have no continuous or equivariant section?
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in ...
- 197
4
votes
0
answers
284
views
Bers' simultaneous uniformization
I have been trying to understand Bers' famous paper "Simultaneous Uniformization". Regarding this paper I have a few questions. Any kind of help will be appreciated.
Let $S$ and $S^{'}$ be two ...
- 221
4
votes
0
answers
613
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
- 1,330
4
votes
0
answers
286
views
Number of connected components of the centre of a Levi subgroup
Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...
- 41
4
votes
0
answers
139
views
Do positive-density subgroups intersect nontrivially?
Let $G$ be an infinite finitely generated group and $S$ a generating set. Define density with respect to the sequence of balls $S^n$.
If $H_1, H_2 \leq G$ have positive density, must $H_1 \cap H_2$ ...
- 9,195
4
votes
0
answers
221
views
Extra line bundles from torsors
Another math.stackexchange question (here: $\mathbb{G}_m$-torsors and line bundles) goes over a way to construct a line bundle $L$ from a $\mathbb{G}_m$-torsor $T \to B$, by using a decomposition $\...
- 909
4
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0
answers
169
views
Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells
I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of ...
- 5,319
4
votes
0
answers
171
views
Borromean Lines of three $\mathbb{R}^1$ in $\mathbb{R}^3$ and analogous Milnor link invariants
It is know that Borromean rings can be detected by Milnor invariant
$$
\bar{\mu}(\gamma_1,\gamma_2,\gamma_3)=
\# (\Sigma_1 \cap \Sigma_2 \cap \Sigma_3)-\frac{1}{2}\sum_{I,J,K}\epsilon_{IJK}
\sum_{\...
- 2,137
4
votes
0
answers
211
views
How frequent are permutations with small interleaving?
For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$.
Let $\pi$ be a permutation of $[n]$. For simplicity, assume that $\pi$ is an $n$-cycle. ...
- 19.4k
4
votes
0
answers
159
views
Trilinear polarity from AG perspective
Consider a triangle $ABC$ in the projective plane $\mathbb{P}^2.$ For a point $p \in \mathbb{P}^2$ one can define its trilinear polar line $t(p)$ (see here). This defines a birational map to the dual ...
- 4,296
4
votes
0
answers
919
views
Guessing of $n$th prime from "super- regularized" product of primes
( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.)
We know "super-...
- 690
4
votes
0
answers
347
views
Mackey theory application - semidirect product abelian-by-finite
In order to advance my research I'm supposed to understand this fact:
Let $A$ be an abelian group and $S$ a finite group acting on $A$.
This defines the semi-direct product $A\rtimes S$:
Let $\chi$ ...
- 51
4
votes
0
answers
247
views
p cohomological dimension of a profinite group
I would like to know what is the $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$. Here $S$ is a finite set of primes containing $p$ and the Archimedean primes and $\mathbb{...
- 1,191
4
votes
0
answers
76
views
Categorical construction of comodule category of FRT algebra
Let $\mathcal{B}$ denote the braid groupoid, with objects being non-negative integers $n \in \mathbb{Z}_{\geq 0}$ and morphisms $\mathcal{B}(n,n)=B_{n}$ given by the braid group. Let $\mathcal{C}$ be ...
- 1,661
4
votes
0
answers
77
views
Proximal isometries in CAT($-1$) metric space
Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...
- 445
4
votes
0
answers
159
views
(Non-)Orientability of non-triangulable manifolds
We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least ...
- 2,137
4
votes
0
answers
41
views
Edge orientation of finite triangle-free graphs
Given a finite simple graph without triangles, I am interested in conditions ensuring that there exists an orientation of the edges such that the following holds.
There exists no cycle $x_0,x_1,\dots,...
- 516
4
votes
0
answers
107
views
Generalized de Rham cohomology on product bundle giving specified cohomology
Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* ...
4
votes
0
answers
68
views
"Robust" Noninjectivity of a Continuous Mapping of a Sphere into the Plane
Let $X=\mathbb{S}^2$ and $Y=\mathbb{R}^2$ and $f:X\to Y$ a continuous mapping. Is it true that there must exist a nonempty set $V\subset f(X)$, open in $f(X)$ (in the subspace topology), such that ...
- 113
4
votes
0
answers
83
views
Algebraic integers whose matrix representations have singular values in an interval
Let $K$ be a finite extension of $\mathbb{Q}$. Let $\mathcal{O}(K)$ be the ring of integers of $K$. Let $\omega_1,\ldots,\omega_n$ be an integral basis for $K$ over $\mathbb{Q}$.
For each $a \in K$...
- 459
4
votes
0
answers
215
views
Properties of triangulations of homeomorphic CW complexes
Let $X$ and $Y$ be two triangulable CW complexes which are homeomorphic.
Is it true that there exists a triangulation of $X$ and a triangulation $Y$ which have a common subdivision?
- 250
4
votes
0
answers
942
views
Next step in studying arithmetic geometry
This relates to this post.
I want to study arithmetic, such as Fermat's last theorem, Faltings' theorem, Mazur's torsion points theorem, Weil conjecture and so on.
For understanding these theorems (...
- 1,352
4
votes
0
answers
362
views
A particular pushout of homologicaly rational spaces
Let $R^{\delta}$ be the topological group of additive real numbers (with discrete) topology and let $R$ be the topological group of additive real number with the standard topology. Let $X$ be a (...
- 1,974
4
votes
0
answers
271
views
What are the normal subgroups of the finite Coxeter Groups of type Bn?
Let $B_n = \langle \rho_0,\rho_1,\ldots,\rho_{n-1} \rangle$ subject to the relations that $(\rho_i\rho_j)^{m_{i,j}} = id$ with $m_{i,i} = 1$, $m_{i,j} = 2$ for $|i-j|\ge 2$, $m_{i,i+1} =3$ for $0 \le ...
- 193